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Given a faithful group action $G$ on a finite set $X$ of cardinality $n$ and an integer $1\le k\le n$, say that $G$ is $k$-set-transitive if we can map any unordered $k$-tuple in $X$ to any other via an element $g\in G$. (Contrast with being $k$-transitive, in which case this holds for ordered tuples.) Say that $G$ is set-transitive if it is $k$-set-transitive for all $k$.

Obviously, $S_n$ and $A_n$ are set-transitive with their natural action on an $n$-element set (except $n=2$ in the alternating case), and in general $k$-transitivity implies both $k$-set-transitivity and $(n-k)$-set-transitivity. The Mathieu group $M_{12}$ is $k$-set-transitive for all $k\neq 6$.

Some questions:

  • Are there any set-transitive actions besides the two categories mentioned above?

  • Besides $A_4$ and $A_5$, are there any groups which are $k$-set-transitive for all $k\le 4$ which are not also $4$-transitive?

  • More generally, are there any group actions which are $k$-set-transitive for some $k\le n/2$ but not $k$-transitive? The condition seems much weaker, but I'm struggling to generate examples.

Any pointers to discussion of this property in the literature would also be welcome.

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The standard term for a group that is transitive on $k$-sets is $k$-homogeneous.

A basic result is due to Livingstone and Wagner (1965) and Kantor (1972):

Suppose that $G \le {\rm Sym}(X)$ with $|X| \le n$, and assume that $G$ is $k$-homogeneous with $k \le n/2$ (that is no restriction since $G$ is $k$-homogeneous if and only if it is $(n-k)$-homogeneous). Then $G$ is $(k-1)$-transitive and, with the following exceptions, $G$ is $k$-transitive:

(i) $k=2$, ${\rm ASL}_1(q) \le G \le {\rm A \Sigma L}_1(q)$ with $q \equiv 3 \bmod 4$;

(ii) $k=3$, ${\rm PSL}_2(q) \le G \le {\rm P \Sigma L}_2(q)$ with $q \equiv 3 \bmod 4$;

(iii) $k=3$, $G={\rm AGL}_1(8)$, ${\rm A \Gamma L}_1(8)$, or ${\rm A \Gamma L}_1(32)$;

(iv) $k=4$, $G={\rm PGL}_2(8)$, ${\rm P \Gamma L}_2(8)$, or ${\rm P \Gamma L}_2(32)$.

This is stated (but not proved) as Theorem 9.4B in Dixon & Mortimer's book "Permutation Groups".

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